STRESSES IN A LAID JOINT WITH ELASTIC ADHESIVE. R A 4,,t. 14o September 1957 FEB
|
|
|
- Andrea Liliana Woods
- 5 years ago
- Views:
Transcription
1 _77,IT: ROOM F6 t STRESSES IN A LAID JOINT WITH ELASTIC ADHESIVE 14o September 1957 R A 4,,t. FEB v.v *PATE Co This Report is One of a Series Issued in Cooperation with the ANC-23 PANEL ON SANDWICH. CONSTRUCTION MR AIRCRAFT of the Departments of the AIR FORCE, NAVY, AND COMMERCE H II rIlm l in.1[111[,' FOREST PRODUCTS LABORATORY MADISON 5, WISCONSIN UNITED STATES DEPARTMENT OF AGRICULTURE FOREST SERVICE In Cooperation with the University of Wisconsin
2 TABLE OF CONTENTS I. SUMMARY 1 II. INTRODUCTION 1 Page III. NOTATION 2 IV. THEORETICAL ANALYSIS 4 V. NUMERICAL COMPUTATIONS 24 VI. CONCLUSIONS 27 VII. REFERENCES 28 VIII. ACKNOWLEDGEMENTS. 29 IX. APPENDICES 30 Appendix A -- Integration Formulas 30 Appendix B -- Euler Transformation for Divergent Series 32 X. FIGURES 33
3 STRESSES IN A LAP JOINT WITH ELASTIC ADHESIVE! By R. E. SHERRER, Engineer Forest Products Laboratory,? Forest Service U. S. Department of Agriculture I. SUMMARY A theoretical solution for the displacements and stresses in the adhesive of a lap joint loaded in tension is presented. The solution is applicable for an anti-symmetrical jointwith arbitrary overlap, adhesive thickness and plate thickness and for adhesive and plates of any arbitrary combination of materials. Elementary theory of bending is used for the plates and theory of elasticity is used for the adhesive. The usual limitations of these theories apply to the solution. Numerical results and curves are included. II. INTRODUCTION The purpose of this report is to solve the following problem: Given two rectangular sheets of any isotropic material and of equal thickness t, of unit width, and of length (/ + 2c) (see figure 1), lap-jointed!this report is one of a series (ANC-23, Item 51-4) prepared and distributed by the U. S. Forest Products Laboratory under U. S. Navy Bureau of Aeronautics Order No. NAer and U. S. Air Force No. AF-18 (600)-102 and DO 33(616) Results here reported are preliminary and may be revised as additional data become available.?maintained at Madison, 'Wis., in cooperation with the University of Wisconsin. Rept. No. 1864
4 2. over a length 2c, bonded by an adhesive layer of any isotropic material and of thickness 2b, and acted upon by tensile forces of magnitude T o per unit sheet width, find the displacements and stresses of all points within the adhesive. For the solution in this report, a free body of the joint section (see figure 2) acted upon by loads as computed in reference 1 will be used. Elementary theory of bending will be used on the rectangular sheets (henceforth called plates) and theory of elasticity, using the Saint-Venant principle on the free edges, will be used for the adhesive. Plane strain is assumed. III. NOTATION x, y rectangular coordinates (figure 1) 2c length of joint 2b thickness of adhesive t thickness of each plate G shear modulus of elasticity for adhesive Poisson' s ratio for adhesive E D G ' Young' s modulus for plates flexural rigidity per unit width of plate 2Gc non-dimensional rigidity factor Et D ' Dnon-dimensional rigidity factor 2Gtc 2 T tension in plate per unit width
5 3. M moment in plate per unit width V transverse shear in plate per unit width To value of T at beginning of joint Mo estimated value of M at beginning of joint T' To, non-dimensional tension factor 4Gc M' Mc, non-dimensional moment factor 12 D Vo V t non-dimensional shear factor 4Gc2 Cr o normal unit stress of adhesive on plate T o tangential unit stress of adhesive on plate v odeflection of plate surface in contact with adhesive u o horizontal displacement of plate surface in contact with adhesive vp deflection of center plane of plate up horizontal displacement of center plane of plate u, v displacements of adhesive o, o-y,, Txy plane stresses in adhesive do cosh a n b en s inh ftn b
6 4. fn anb sinh anb - (1-2p.) cosh anb gn f3n b cosh Pn b - (1-2p.) sinh f3nb hn anb sinh an b - (3-4p.) cosh anb in Pn b cosh (an b - (3-4p.) sinh Pnb i n anb cosh an b - 2(1 - p.) sinh anb kn sinh an b fn an b cosh anb mn Pn b sinh 13n b nncosh On b on grib sinh Pnb - 2(1 - IA) cosh Pnb IV. THEORETICAL ANALYSIS To simplify the writing, the units of length will be taken such as to make c equal to unity. This will have the effect of making b, t, u, v, uo, vo, u, v and the coordinates x and y non-dimensional, since b p p will be written for b/c, t for t/c, etc. From the free-body diagram shown in figure 3 a- dv 0 o - dx.t.0 _ dt 0 dv dm 2 dx dx 0
7 5. where the moment M is given by M = D d 2 v, (4) Elimination of V and M gives d4v dr_ dv. D dx4 P + Cr = dx (T ) (5) as the differential equation for the plates. It will be assumed that the displacements of the central plane of the upper plate are related to the displacements of the lower surface of the plate as follows: v" = vo t dvo u p = u oz dx These assumptions are in accord with the elementary theory of bending used for the plates. The unit strain at the central plane of the plate is given by dup T dx Et (8) From equations (7) and (8) T duo t d2vo = 0 (9) Et dx 2 dx Equations (5) and (9) are the two differential equations to be solved. It
8 6. is noted that the term on the right side of equation (5) is non-linear since both T and vpdepend on the applied load. The differential equations for the adhesive based upon plane strain are I 8 au av 1-2p. ax ) "2u = (10) 1 8,8u, Ov% 1 v-v = 0 1-2p. 8y 8x 8y The boundary conditions for the upper plate are (see figure 2) T = T o at x = 1 (12) T = 0 at x = -1 (13) M = Mo at x = 1 (14) M = 0 at x = -1 (15) V = -Vo at x = 1 (16) V = 0 at x = -1 (17) To simplify the analysis, the principle of Saint-Venant will be used for the free boundaries (x = ±1) of the adhesive. This imposes the following conditions: crxdy = 0 (18) x = ±1
9 7. -b 6x y dy = 0 (19) x = ±1 T xy x = *1 dy = 0 (20) -b The conditions at the boundary between the adhesive and the upper plate will be assumed to be Oro = Cry at y = b (21) To = Tx at y = b (22) u 0 = u at y = b (23) vo = v at y = b (24) Similar conditions will apply for the lower plate, but these do not have to be expressed if the necessary conditions of symmetry and anti-symmetry are observed when writing the expressions for displacements. If any term in the expressions for u and v is even in x, it must be odd in y and vice versa. Expressions for u and v satisfying equations (10) and (11) and with sufficient arbitrary constants to satisfy all boundary conditions as well as equations (5) and (9) will now be written. The details of how the
10 8. various terms of these expressions were obtained is not important and will not be given. That each term does satisfy the differential equations (10) and (11) may be verified by substitution. Equations (12). to (20) give nine conditions for which nine arbitrary constants (a 1 to a 9 ) have been provided. The first seven of these are used in the expressions for u and v, the eighth comes from the integration of equation (2) and the ninth from integration of equation (1). The plate differential equations (5) and (9) together with conditions at the boundary between the plate and adhesive make it necessary to add to the expressions for u and v terms which become Fourier series for this boundary. The ex- _ pressions follow: d [ 3 u = a y + a 2 (1 - µ)xx + a 3xy2 a (3. 3 x2y x 5 + (1-2p.)x 3y 2 - (1 - p.)xy4 - a6 4x y a (1-2p.) x 4 y - (1 - p.)x2y 3 + (3-214 y Anorny cosh a y n + A' Binh a y n n=1 YE nf3ny sinh 13ny + Bn cosh 13n1 n=1 cos ax an sin Pnx (25)
11 9. 0 t;, Is) ta V )4.5 t 03 )4 I o al 0 I il I " / c 1. 1 I f NI 1 er...0.% 111 ir.'"..'''... 1 in. 0 cd a) ti il 5 + ti 1 0 to r="1 en r r--1 t u. ) CCI M 'Cr 4 I ± i... I m i u + en.--- v I N 14 r.1 i a 0 cd14 M)4 I PI 1 r 1 er= ccl I.. ( i i s tl C in to,.. m o ini.... to t) A1_2_1 ± > N 0 0 m t03 al 0 1 cni -4 fil SwEiNie.ao!" ± rd N [ id I + I ii
12 10 ti N k71 i il r4 r..._ i N r7-4-1 N, N a I....-j, en + M - X d e cl "--"..=/ CCL CCL ZS A N A I >4 d 1 I...v..., tn o U 4 ul u MI 1:q d d MI e+.) Tr.. W 0 Lt' Lai
13 11. r ev > I co '-. 3- >4 1 1 NI 0 CCL ti to NI,------""---t f--..";:---.1 IN I ' 0 >4 Zs 0 Cia, 3. M r*t I -0, -0 LI) 0 1 x.5 u rn.44 i. i 171 r-al 4 PI t4 + r.3 v a '2 + I I 1...,:,.. j en,,4...4 td X IN3 IN 1... i I I N X id *-$ ,;± j 4.2._ 7.. NI I ren i X, O. '447 ti c to I tfi, COI CE, I O 0:1 NI + ^., pri tl ca COI cr3 X 0,..., c. i i N I M en cd 1/4...2,.._ 4 >4 3. _ ' l f1:1 cli. ---J 'I Li 1d 0 0 I tn r., I.0 0 N 0 ' N b.
14 12. I p en M >4 1 m o ti CO_ ml U) g N0...4 U co >4 af '.-----Th r >4 1 N ti Ca. M 1 to I to + ti rr--, N 14 >4 0 il + " fll _I m N. a N cd " J. I I + ct) > I L-0 )4 Cd Lfl ni > --;' I, M en. rd.0 s i Nt+ l.41 1 ev, mi x 7. ± cr NI -0 o I o - < i t P r g1:1 1..._1 + o C 1 NI,.0 so..-., 0 en en c" i,, NtIm 0 0 tto C v14 <4 1- Nr_a CI:1 L...Ne--,-II en I co p,_ rd N 0 N O mi '[. 91 >- C 10
15 13. From equations (2), (22), and (29) 1 T = ZG si x 3a 3 bx2 - a4 x 3 a 5 l i bx4 + (1 - p.)b3x2 2 3(1 - p,) 2 - [2-11 ) 3,-5 _ w 20 (1 µ) b 2x 3 _ x + a "I µx 5 (1 - /1) b2x3 2 4 ' 20 2 (2 -µ)o] + a 8 + ZG A n f n + A n ' d 4 n=1 sin anx n - 2G [3ngn + Bn e (30) n=1 From equations (4), (6) and (26) d CO5 finx Pn M a 3b + a4 2(2 - /1) x + 6a. 5 bx 2-9- a 6 ( 1. - (3-2µ) x3-6(1 - ix) b 2x 1 - a 7 x 3) D [Anhn n=1 + An an sin anx + DBin +B' n el n n=1 [n From equations (1), (21) and (28) gii cos Onx (31)
![14. V = - 2G a 9 - a (x 3-3 3b2x) + a 4 (1 b x2 - p.) a 5 + ) x5 p.b 2x 3 - (1-2 b4] + a 6 ( 2 - /1) bx4 4 (1 - p.) b 3x2 a a bx4 4.](/docs-images/92/108108965/images/16-1.jpg "(1 1-4 3x + 2G 2 7 4 2 n=1 + An kd c 13 anx an sin 13nx + 2GZ E3n o n + B ' n n n=1 Pn (32) It is advantageous to select an and On such that siriai = 0 (33) cos Sri 0 (34) These relations will be")
16 14. V = - 2G a 9 - a (x 3-3 3b2x) + a 4 (1 b x2 - p.) a 5 + ) x5 p.b 2x 3 - (1-2 b4] + a 6 ( 2 - /1) bx4 4 (1 - p.) b 3x2 a a bx4 4. ( x + 2G n=1 + An kd c 13 anx an sin 13nx + 2GZ E3n o n + B ' n n n=1 Pn (32) It is advantageous to select an and On such that siriai = 0 (33) cos Sri 0 (34) These relations will be used wherever they apply. The boundary conditions, equations (12) to (20), will now be used to obtain nine equations between the constants a l to a 9 inclusive. From equations (12), (13) and (30) - 3a b -a E b+ (1 - p)bl + a 8 = T' (35)
17 15. 1a4 a - - a6 ( _ 114 l 2 3(1 - p.) k-. + (1aa) b 2 _ (2 - ii) b4 = T' (36) From equations (14), (15) and (31) a 3 + a 5 = - M' b (37) a4 2(2 - P. ) + a6 (3-2p.) - 6(1 - lob ] - a7 = - 6M' (38) (1 - p.) From equations (16), (17) and (32) a 9 + a. a6 - (]L - b 3a7 Eb ) b3 4 (1 - p.) [Awl n + A'. 1(11 cos an - V' (39) an n=1 a 3 (1-3b 2 ) + a 5 (1 + /1) 10 (1-1.0d 2 Y + B n o n + 13, n n n n=1 sin gn Pn = - V (40)
![From equations (18) and (27) 16. a2 + a 3 (3 - b 2) + a 5 + (1 - p.)b 2- IL:Lk) b4 = 0 (41) 10 From equations (19) and (27) a (1-11) µb 6 a - 11 ) _ (2 - b].](/docs-images/92/108108965/images/18-1.jpg "(42) 3 5 3 5 From equations (20) and (29) al 2 ae2 _a4 6 - (1-1.1.) b2 1-11) 4 2 20 b (1 - p) a 7 E b2 4 2 (2 P.")
18 From equations (18) and (27) 16. a2 + a 3 (3 - b 2) + a 5 + (1 - p.)b 2- IL:Lk) b4 = 0 (41) 10 From equations (19) and (27) a (1-11) µb 6 a - 11 ) _ (2 - b]. (42) From equations (20) and (29) al 2 ae2 _a4 6 - (1-1.1.) b2 1-11) b (1 - p) a 7 E b2 4 2 (2 P.)b] 20 + )-F An j n n=i cos a + A n' kri a n b n = 0 (43) Equations (5) and (9) provide the additional relations that are required. From equation (9)
![= 0 n (44) From equation (5) Ho + H1 x+ H 2x 2 + H 3x 3 + H4x 4 - thnon3 D ' 2 j n + a n f n] An + d n](/docs-images/92/108108965/images/19-2.jpg "cr 3 D' + --k n n =1 + andd All sin a x n=1 + [ e ni3n D' + 2 n t n + 0 n e n 13 n 1 ni3n3 D + 2 on -")
19 F 0 +F 1 x +F 2 x 2 + F 3 x 3 + F4 x 4 + F 5 x Y n=1 hat f - An + kn dncrnt drip' + 2 A in} sin anx nt gn1 En inp 2 n=1 Pn nn enpnt 2 eng1 8 } cos O nx = 0 n (44) From equation (5) Ho + H1 x+ H 2x 2 + H 3x 3 + H4x 4 - thnon3 D ' 2 j n + a n f n] An + d n cr 3 D' + --k n n =1 + andd All sin a x n=1 + [ e ni3n D' + 2 n t n + 0 n e n 13 n 1 ni3n3 D + 2 on - fingd Bn 1 d dvn cos Prix * _ --(T--'-.) Gt clx (ix (45)
![18. Where F0 = - a 2 (1 - + 3a 3b [b + a 5(1 - p.)b4 + a 8 G ' (46) F1-1 al G + a4 211 b (1 _ (2 - ti)] - a6 3(1 - p.)b2t [ + E b4g 4 2(1-1.1.)b 3 + (2 - FL) b1/41 (47) 4 F 2 = - + b a 5 3(1 =- 2p.](/docs-images/92/108108965/images/20-1.jpg ")b 2 + 3bt + (1 - p.)b 3G (48) F = a G + a 6 b ±(3 3 4 3(1 - p,) 2-2µ)t+ 1. (1 Ii)b 2 G ' 2 + a E _ 2p.")
20 18. Where F0 = - a 2 ( a 3b [b + a 5(1 - p.)b4 + a 8 G ' (46) F1-1 al G + a4 211 b (1 _ (2 - ti)] - a6 3(1 - p.)b2t [ + E b4g 4 2(1-1.1.)b 3 + (2 - FL) b1/41 (47) 4 F 2 = - + b a 5 3(1 =- 2p.)b 2 + 3bt + (1 - p.)b 3G (48) F = a G + a 6 b ±( (1 - p,) 2-2µ)t+ 1. (1 Ii)b 2 G ' 2 + a E _ 2p.) b + + Ob2G] (49) 7 2 ae + L a be' 2 (50) F 5 = 6 (2 - G + a 7 -L G' (51) H o = [32 + a 5 libd' 3 t _ ( (1 - p.)b1(52) t
![19. H i = - a 4b 2 + a6 6(3 - zon (1 - (1 - + LL1.0.1,3 + 3(1 - }L)b 2 - a 7 6D' - 2 (1 11 ) b 3-3(1 - YID 2 (53) H 2 a _6a 3 t 5 t 61.1.1] (54) H 3 = - a6 2(2 - + (2-4) + a 7-21 1D + t t (55) H 4 = a 5 (1 t (56) Relations for the constants, A A ' B and B, can be found by n ' n applying integral transforms to equations (44) and (45).](/docs-images/92/108108965/images/21-2.jpg "Using the sine transform (see Appendix A) on equation (44) gives 2 cos an F l 4 a6 ) F3 + _ 20 + 120 ) F5 an a n [ n h n a n t InG ' dnant cing = 0 An [cn a 2 + a A n 2 n (57)")
21 19. H i = - a 4b 2 + a6 6(3 - zon (1 - (1 - + LL1.0.1,3 + 3(1 - }L)b 2 - a 7 6D' - 2 (1 11 ) b 3-3(1 - YID 2 (53) H 2 a _6a 3 t 5 t ] (54) H 3 = - a6 2(2 - + (2-4) + a D + t t (55) H 4 = a 5 (1 t (56) Relations for the constants, A A ' B and B, can be found by n ' n applying integral transforms to equations (44) and (45). Using the sine transform (see Appendix A) on equation (44) gives 2 cos an F l 4 a6 ) F3 + _ ) F5 an a n [ n h n a n t InG ' dnant cing = 0 An [cn a 2 + a A n 2 n (57)
22 20. The cosine transform on equation (44) gives 2 sin /3 n F0, + kl - 2 ) F2 + Pn On On 24% 74/ Pn F4 - mn n13nt gngl 2 13n B n en(3nt 2 e /3n B' = 0 (58) n The sine transform on equation (45) gives 2 cos an an H 1 + ( a 3 D' n n + jn + a f] A n n n + d a3 13' + 2 k + a d A' = P n n n t n n n n (59) The cosine transform on equation (45) gives 2 sin I3n H o + (1 - ) H i 133 D pz (1 - p2 ) H p4 4 n n + 2 o + 13 in nn g 1 B + n 3 2 en fin D I + i- nn + e n n B' = Qn n where 1 d (T dv v P = 1 ) 2Gt...1 dx dx sin anx dx (61) (60 )
23 Q n 1 1 f dv d (T ) cos Pnclx 2Gt clx dx (62) Equations (35) to (43) and (57) to (60) form a matrix from which the constants may be solved. A simplification of the matrix may be made by multiplying equation (43) by b and adding the result to equation (39). The resulting equation, b a _Eb 5 - a7 ( = a 9 V + al (63) will replace equation (39). The nature of the matrix is such that it may be broken into two independent matrices. By using the aforementioned equations, together with equations (46) to (56), matrices (64) and (65) are formed. Solving these matrices (with a finite number of equations) for the constants and substituting the constants into equations (25) to (29) will give the displacements and stresses for the adhesive.
24
25
26 24. V. NUMERICAL COMPUTATIONS Mo and Vo are computed from a given force T o by means of the following equations: Mo =k b) T 2 2 (66) = k (t 2b) (67) where the values of k and k' are taken from table 1 of reference 1. In order to simplify the computations, the non-linear terms, P n and Qn, of the matrices, were set equal to zero. Reference to equations (61) and (62) together with equations (26) and (30) will show that each term of P n and Qn involves the multiplication of two small constants thereby making the contribution of Pn and Qn to the matrix significantly small. The fifth equation of matrix (64) and the fourth equation of matrix (65) each contain a divergent series. Both of these equations are used to satisfy the shear stresses on the boundaries of the adhesive and plates. Since they describe a physical condition of the problem, they must sum to a finite value. The problem then is to determine the values of the summations with a finite number of terms. There are several methods of summing divergent series, but the one best suited for this problem is the Euler transformation for divergent series (see
27 Appendix B). The reasons are that in both series the terms in general alternate in sign, remain fairly constant in magnitude (for large values 1 of _c ) for matrix (64), and decrease nearly as for matrix (65). These are conditions for which the Euler transformation works very well. Since one of the divergent series comes from equation (29) for shear stress, it will be necessary to treat both equations the same way in order to satisfy the boundary condition for shear. Hence, if the Euler transform is used in the matrix, it must be used on equation (29). For the other stresses and the displacements it is not so easy to determine whether to use the Euler transformation or not. If a series already converges iri a satisfactorily way, the application of the Euler transformation may slow the convergence or even cause the series to diverge. Since it is difficult to determine the need of the Euler transformation analytical, the following three cases were studied numerically. (a ) Transformation is not used on any equation (b ) Euler transformation used on matrix equations only (c Euler transformation used on matrices, displacement and stress equations. A typical comparison of u, v, crx, o-, and rxy is shown in figures (4) to (8). A study of these curves and other similar comparison curves led to the following conclusions: 25.
28 2 6. (1) use the Euler transformation on the matrices (2) do not use the Euler transformation on u (3) the same results are obtained with or without the Euler trans - formation on v (4) use the Euler transformation on cr x (5) the same results are obtained with or without the Euler transformation on cr (6) use the Euler transformation on T xy It is possible that different choices of material and different ranges of dimensions would lead to different conclusions. Also, some other transformation may work better than the Euler transformation. Having established the numerical procedure, it remains to determine how many Fourier terms are needed to give satisfactory answers. Figures (9) to (13) show the effects of adding these terms to the solution. From a study of these curves and other similar curves, the following conclusions may be drawn. (1) displacement u converges in a satisfactory way, but will require several Fourier terms (2) displacement v could not be made to converge by any method tried (3) stress Tx could not be made to converge by any method tried
29 27. (4) the maximum value of stress cr is obtained with very few Fourier terms, but the distribution will not be established with these terms (5) the shear stress T distribution could not be established with xy the number of terms taken. The most important stress in the design of lap joints is the normal stress cr y, as a study of the stress curves will verify. The maximum normal stress is well established with y = 4. Therefore, a study of this normal stress is made under varying parameters. Figure (14) shows the effect of varying the glue thickness while holding all other variables constant. Likewise, figure (15) shows the effect of varying the plate thickness and figure (16) shows the effect of varying the load. VI. CONCLUSIONS For the most important stress Ly, reliable computations can be made from the solution. This is also true for the displacement u. The shear stress.rxy requires further study and it appears that all that is required is to take many more Fourier terms. For the displacement v and the normal stress o- x some means of obtaining convergence has to be found.
30 28. VII. REFERENCES (1) Goland, M. and Reissner, E Stresses in Cemented Joints. Journal of Applied Mechanics, Vol. II. (2) Kuenzi, E. W Determination of Mechanical Properties of Adhesives for Use in the Design of Bonded Joints. Forest Products Laboratory Rept. No
31 2 9. VIII. ACKNOWLEDGEMENTS The author is indebted to the Forest Products Laboratory and to the Engineering Experimental Station of the University of Wisconsin for providing financial support for this project. The author wishes to thank Prof. Gerald Pickett of the Mechanics Department of the University of Wisconsin and Mr. Charles Norris of the Forest Products Laboratory for their advice, guidance and encouragement throughout the course of this work.
32 30. IX. APPENDICES 1. Appendix A -- Integration Formulas Integration formulas used in this report include the following: sin anx dx = 0 (48) x sin a nx dx = 2 cos a n (69) x 2 sin a x dx = n 0 (70) S- 4. x 3 sin a x dx n 2 cos a n (1 -) (71) an 4 x sin a n x dx = 0 (72) 5 s. x m ax dx n 2 cos a n ki - 1 (73) an a n a4 n
33 cos g n x dx = 2 sin 13 n n (74) x cos Pnx dx '= 0 (75) 1 x 2 cos gnx dx = 2 Bin gn 1311 (76) x 3 cos gnx dx = 0 (77) x4 cos gnx dx 2 sin /3 n on4 (78) x 5 cos Pax dx = 0 (79)
34 Appendix B -- Euler Transformation for Divergent Series The transformation is usually written as follows: a l a l + a 2 a a + 2a + a a 2+ aa + 3a + 3a + a z 3 4 a 3a 2 + 2a + a 3 4 a + a 3 4 a 4 (80) where al, a2, a 3, etc. are terms of the original series. The new series is formed as follows: 1 1 a + (a az + (a 1 + 2a2 + a 3) + (a + 3a a 3 + a 4 ) +.. (81) In the series used in this report, the identity of the original terms has to be maintained since the terms are used in amatrix. In order to do this the series is rearranged in the following manner Kl a l + K 2a2 + K 3a Knan + + Kta. (83) k where (m - 1)! K n (83) ni=.n (rn - n)! (n - 1)1 2m and k is the total number of terms used in the series.
35
36
37
38
39
40
41
42
43
44
45
46
COMPRESSIVE EUCICLING CURVES MR SANDWICH PANELS WITH ISOTROPIC FACINGS AND ISOTROPIC OR ORTI1OTROIPIC CORES. No Revised January 1958
SRICULTU RE ROOM I COMPRESSIVE EUCICLING CURVES MR SANDWICH PANELS WITH ISOTROPIC FACINGS AND ISOTROPIC OR ORTI1OTROIPIC CORES No 1854 Revised January 1958 This Report is One of a Series Issued in Cooperation
ri [11111 IlL DIRECTIONAL PROPERTIES Of GLASS-FABRIC-BASE PLASTIC LAMINATE PANELS Of SIZES THAT DO NOT IBUCICLE (P-Q1lAtVjr) No.
Supplement to DIRECTIONAL PROPERTIES Of GLASS-FABRIC-BASE PLASTIC LAMINATE PANELS Of SIZES THAT DO NOT IBUCICLE (P-Q1lAtVjr) No. 1803-13 November 1955 This Report is One of a Series issued hi Cooperation
TORSION OF SANDWICH PANELS OF TRAPEZOIDAL, TPIANGUIAP, AND RECTANGULAR CROSS SECTIONS
!CULTURE ROOM TORSION OF SANDWICH PANELS OF TRAPEZOIDAL, TPIANGUIAP, AND RECTANGULAR CROSS SECTIONS June 1960 No. 1874 This Report Is One of a Series Issued in Cooperation with the ANC-23 PANEL ON COMPOSITE
(laod No \' V,R A " FI- 1 4, <4. ELASTIC STABILITY Of CYLINDRICAL SANDWICH SHELLS UNDER AXIAL AND LATERAL LOAD. July 1955
ELASTIC STABILITY Of CYLIDRICAL SADWICH SHELLS UDER AXIAL AD LATERAL LOAD (laod o. 1852 July 1955 This Report is One of a Series Issued ha Cooperation with the AC-23 PAEL O SADWICH COSTRUCTIO of the Departments
BUCKLING COEFFICIENTS FOR SIMPLY SUPPORTED, FLAT, RECTANGULAR SANDWICH PANELS UNDER BIAXIAL COMPRESSION
U. S. FOREST SERVICE RESEARCH PAPER FPL 135 APRIL 1970 BUCKLING COEFFICIENTS FOR SIMPLY SUPPORTED, FLAT, RECTANGULAR SANDWICH PANELS UNDER BIAXIAL COMPRESSION FOREST PRODUCTS LABORATORY, FOREST SERVICE
MINIMUM WEIGHT STRUCTURAL SANDWICH
U.S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY MADISON, WIS. In Cooperation with the University of Wisconsin U.S.D.A. FOREST SERVICE RESEARCH NOTE Revised NOVEMBER 1970 MINIMUM
ELASTIC STAIBILITY CIF TUE FACINGS Of HAT SANDWICI-1 PANELS WIASI SUBJECTED TO COMBINED EDGEWISE STRESSES
ELASTIC STAIBILITY CIF TUE FACINGS Of HAT SANDWICI-1 PANELS WIASI SUBJECTED TO COMBINED EDGEWISE STRESSES Information Reviewed and Reaffirmed Aucust 1955 NFORMA-tiON RE'4,E\AE.'L; n PE.1-17;9';f2,. This!Report
STRESSES WITHIN CURVED LAMINATED BEAMS OF DOUGLAS-FIR
UNITED STATES DEPARTMENT OF AGRICULTURE. FOREST SERVICE - FOREST PRODUCTS LABORATORY - MADISON, WIS. STRESSES WITHIN CURVED LAMINATED BEAMS OF DOUGLAS-FIR NOVEMBER 1963 FPL-020 STRESSES WITHIN CURVED LAMINATED
PLASTIC FLOW THROUGHOUT VOLUME OF THIN ADHESIVE!BONDS. No March 1958 (.2. Will In' iriculture ROOM. Mum mina
iriculture ROOM (.2 PLASTIC FLOW THROUGHOUT VOLUME OF THIN ADHESIVE!BONDS No. 2092 March 1958 11 Will In' Mum mina FOREST PRODUCTS LABORATORY MADISON 5, WISCONSIN UNITED STATES DEPARTMENT OF AGRICULTURE
ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY THE SUPERPOSITION METHOD
Journal of Sound and Vibration (1999) 219(2), 265 277 Article No. jsvi.1998.1874, available online at http://www.idealibrary.com.on ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY
Mechanics in Energy Resources Engineering - Chapter 5 Stresses in Beams (Basic topics)
Week 7, 14 March Mechanics in Energy Resources Engineering - Chapter 5 Stresses in Beams (Basic topics) Ki-Bok Min, PhD Assistant Professor Energy Resources Engineering i Seoul National University Shear
Laminated Beams of Isotropic or Orthotropic Materials Subjected to Temperature Change
United States Department of Agriculture Forest Service Forest Products Laboratory Research Paper FPL 375 June 1980 Laminated Beams of Isotropic or Orthotropic Materials Subjected to Temperature Change
UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich
UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST
I9 Lateral deflections of circular plates
I9 Lateral deflections of circular plates 19.1 Introduction In this chapter, consideration will be made of three classes of plate problem, namely (i) (ii) (iii) small deflections ofplates, where the maximum
WRINI CLING Of THE FACINGS OF SANDWICH CONSTRUCTION %EJECTED TO EDGEWISE COMPRESSION Sandwich Constructions Having Honeycomb Cores
WRINI CLING Of THE FACINGS OF SANDWICH CONSTRUCTION %EJECTED TO EDGEWISE COMPRESSION Sandwich Constructions Having Honeycomb Cores June 1953 This Report is One of a Series Issued in Cooperation with the
APPENDIX A Thickness of Base Metal
APPENDIX A Thickness of Base Metal For uncoated steel sheets, the thickness of the base metal is listed in Table A.1. For galvanized steel sheets, the thickness of the base metal can be obtained by subtracting
Accordingly, the nominal section strength [resistance] for initiation of yielding is calculated by using Equation C-C3.1.
![Accordingly, the nominal section strength [resistance] for initiation of yielding is calculated by using Equation C-C3.1.](/thumbs/89/98617066.jpg "Accordingly, the nominal section strength [resistance] for initiation of yielding is calculated by using Equation C-C3.1.")
C3 Flexural Members C3.1 Bending The nominal flexural strength [moment resistance], Mn, shall be the smallest of the values calculated for the limit states of yielding, lateral-torsional buckling and distortional
Mechanics of Materials II. Chapter III. A review of the fundamental formulation of stress, strain, and deflection
Mechanics of Materials II Chapter III A review of the fundamental formulation of stress, strain, and deflection Outline Introduction Assumtions and limitations Axial loading Torsion of circular shafts
MECHANICS OF MATERIALS
STATICS AND MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr, John T. DeWolf David E Mazurek \Cawect Mc / iur/» Craw SugomcT Hilt Introduction 1 1.1 What is Mechanics? 2 1.2 Fundamental
COMPOSITE PLATE THEORIES
CHAPTER2 COMPOSITE PLATE THEORIES 2.1 GENERAL Analysis of composite plates is usually done based on one of the following the ries. 1. Equivalent single-layer theories a. Classical laminate theory b. Shear
3. BEAMS: STRAIN, STRESS, DEFLECTIONS
3. BEAMS: STRAIN, STRESS, DEFLECTIONS The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets
FLEXURE OF STRUCTURAL SANDWICH CONSTRUCTION
FLEXURE OF STRUCTURAL SANDWICH CONSTRUCTION December 1951 INFORMATION REVIEWED AND REAFFIRMED 1958 LOAN COPY Please return to: Wood Engineering Research Forest Products Laboratory Madison, Wisconsin 53705
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 11 Last class, what we did is, we looked at a method called superposition
Parts Manual. EPIC II Critical Care Bed REF 2031
EPIC II Critical Care Bed REF 2031 Parts Manual For parts or technical assistance call: USA: 1-800-327-0770 2013/05 B.0 2031-109-006 REV B www.stryker.com Table of Contents English Product Labels... 4
3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
Aircraft Structures Kirchhoff-Love Plates
University of Liège erospace & Mechanical Engineering ircraft Structures Kirchhoff-Love Plates Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/ Chemin
Level 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method
9210-203 Level 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method You should have the following for this examination one answer book No additional data is attached
Name :. Roll No. :... Invigilator s Signature :.. CS/B.TECH (CE-NEW)/SEM-3/CE-301/ SOLID MECHANICS
Name :. Roll No. :..... Invigilator s Signature :.. 2011 SOLID MECHANICS Time Allotted : 3 Hours Full Marks : 70 The figures in the margin indicate full marks. Candidates are required to give their answers
EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading
MA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading MA 3702 Mechanics & Materials Science Zhe Cheng (2018) 2 Stress & Strain - Axial Loading Statics
General elastic beam with an elastic foundation
General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation
Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship
Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction
FIXED BEAMS IN BENDING
FIXED BEAMS IN BENDING INTRODUCTION Fixed or built-in beams are commonly used in building construction because they possess high rigidity in comparison to simply supported beams. When a simply supported
Nomenclature. Length of the panel between the supports. Width of the panel between the supports/ width of the beam
omenclature a b c f h Length of the panel between the supports Width of the panel between the supports/ width of the beam Sandwich beam/ panel core thickness Thickness of the panel face sheet Sandwich
Stresses in Curved Beam
Stresses in Curved Beam Consider a curved beam subjected to bending moment M b as shown in the figure. The distribution of stress in curved flexural member is determined by using the following assumptions:
Due Monday, September 14 th, 12:00 midnight
Due Monday, September 14 th, 1: midnight This homework is considering the analysis of plane and space (3D) trusses as discussed in class. A list of MatLab programs that were discussed in class is provided
CIV-E1060 Engineering Computation and Simulation Examination, December 12, 2017 / Niiranen
CIV-E16 Engineering Computation and Simulation Examination, December 12, 217 / Niiranen This examination consists of 3 problems rated by the standard scale 1...6. Problem 1 Let us consider a long and tall
CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES
CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES 14.1 GENERAL REMARKS In structures where dominant loading is usually static, the most common cause of the collapse is a buckling failure. Buckling may
INTRODUCTION TO STRAIN
SIMPLE STRAIN INTRODUCTION TO STRAIN In general terms, Strain is a geometric quantity that measures the deformation of a body. There are two types of strain: normal strain: characterizes dimensional changes,
(MPa) compute (a) The traction vector acting on an internal material plane with normal n ( e1 e
EN10: Continuum Mechanics Homework : Kinetics Due 1:00 noon Friday February 4th School of Engineering Brown University 1. For the Cauchy stress tensor with components 100 5 50 0 00 (MPa) compute (a) The
STATICALLY INDETERMINATE STRUCTURES
STATICALLY INDETERMINATE STRUCTURES INTRODUCTION Generally the trusses are supported on (i) a hinged support and (ii) a roller support. The reaction components of a hinged support are two (in horizontal
Basic Energy Principles in Stiffness Analysis
Basic Energy Principles in Stiffness Analysis Stress-Strain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting
Lecture 15 Strain and stress in beams
Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME
Problem " Â F y = 0. ) R A + 2R B + R C = 200 kn ) 2R A + 2R B = 200 kn [using symmetry R A = R C ] ) R A + R B = 100 kn
![Problem  F y = 0. ) R A + 2R B + R C = 200 kn ) 2R A + 2R B = 200 kn [using symmetry R A = R C ] ) R A + R B = 100 kn](/thumbs/88/116790835.jpg "Problem  F y = 0. ) R A + 2R B + R C = 200 kn ) 2R A + 2R B = 200 kn [using symmetry R A = R C ] ) R A + R B = 100 kn")
Problem 0. Three cables are attached as shown. Determine the reactions in the supports. Assume R B as redundant. Also, L AD L CD cos 60 m m. uation of uilibrium: + " Â F y 0 ) R A cos 60 + R B + R C cos
Cyby TISIA E. Document No. - "/r ".. TECHNICAL LIBRARY /, Copy No. /- ' E. E. Secher. 2 uut15. No. EM JReport
.All I( l-, ltc I+ / '" TECHNICAL LIBRARY /, Document No. - "/r ".. Copy No. /- ' Cyby E. E. Secher " ~~ ~ ~~ A I P"l +" ".i JReport No. EM 9-18 2 uut15 H.ARD MIfROFICHE COPY $o 6. - S PA CE 1NO TEC- LO.aQG3Y
IDE 110 Mechanics of Materials Spring 2006 Final Examination FOR GRADING ONLY
Spring 2006 Final Examination STUDENT S NAME (please print) STUDENT S SIGNATURE STUDENT NUMBER IDE 110 CLASS SECTION INSTRUCTOR S NAME Do not turn this page until instructed to start. Write your name on
Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method
Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method Objectives In this course you will learn the following Deflection by strain energy method. Evaluation of strain energy in member under
UNIT III DEFLECTION OF BEAMS 1. What are the methods for finding out the slope and deflection at a section? The important methods used for finding out the slope and deflection at a section in a loaded
CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES
CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES * Governing equations in beam and plate bending ** Solution by superposition 1.1 From Beam Bending to Plate Bending 1.2 Governing Equations For Symmetric
CHAPTER 5. Beam Theory
CHPTER 5. Beam Theory SangJoon Shin School of Mechanical and erospace Engineering Seoul National University ctive eroelasticity and Rotorcraft Lab. 5. The Euler-Bernoulli assumptions One of its dimensions
EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE STRESS DISTRIBUTION IN PLATES
EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE STRESS DISTRIBUTION IN PLATES OF WOOD OR PLYWOOD CONSIDERED AS ORTHOTROPIC MATERIALS Information Revied and Reaffirmed March 1956 No. 1510 EFFECT OF ELLIPTIC
UNIT- I Thin plate theory, Structural Instability:
UNIT- I Thin plate theory, Structural Instability: Analysis of thin rectangular plates subject to bending, twisting, distributed transverse load, combined bending and in-plane loading Thin plates having
Applications of the Plate Membrane Theory
Chapter 2 Applications of the Plate Membrane Theory In this chapter we will give solutions for plates, which are loaded only on their edges. This implies that no distributed forces p x and p y occur, and
ε t increases from the compressioncontrolled Figure 9.15: Adjusted interaction diagram
CHAPTER NINE COLUMNS 4 b. The modified axial strength in compression is reduced to account for accidental eccentricity. The magnitude of axial force evaluated in step (a) is multiplied by 0.80 in case
Submitted to the Highway Research Board for. Publication in, the lfhighway Research Record!! CONTRIBUTION OF LONGITUDINAL STIFFENERS OF PLATE GIRDERS
Submitted to the Highway Research Board for Publication in, the lfhighway Research Record!! THE CONTRIBUTION OF LONGITUDINAL STIFFENERS TO THE STATIC STRENGTH OF PLATE GIRDERS by Peter B. Cooper Lehigh
Discontinuous Distributions in Mechanics of Materials
Discontinuous Distributions in Mechanics of Materials J.E. Akin, Rice University 1. Introduction The study of the mechanics of materials continues to change slowly. The student needs to learn about software
Using the finite element method of structural analysis, determine displacements at nodes 1 and 2.
Question 1 A pin-jointed plane frame, shown in Figure Q1, is fixed to rigid supports at nodes and 4 to prevent their nodal displacements. The frame is loaded at nodes 1 and by a horizontal and a vertical
Higher Order Linear Equations
C H A P T E R 4 Higher Order Linear Equations 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function g(t) = t, are continuous everywhere. Hence solutions are valid
[8] Bending and Shear Loading of Beams
![[8] Bending and Shear Loading of Beams](/thumbs/92/110949676.jpg "[8] Bending and Shear Loading of Beams")
[8] Bending and Shear Loading of Beams Page 1 of 28 [8] Bending and Shear Loading of Beams [8.1] Bending of Beams (will not be covered in class) [8.2] Bending Strain and Stress [8.3] Shear in Straight
PURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC.
BENDING STRESS The effect of a bending moment applied to a cross-section of a beam is to induce a state of stress across that section. These stresses are known as bending stresses and they act normally
ELASTICITY AND FRACTURE MECHANICS. Vijay G. Ukadgaonker
THEORY OF ELASTICITY AND FRACTURE MECHANICS y x Vijay G. Ukadgaonker Theory of Elasticity and Fracture Mechanics VIJAY G. UKADGAONKER Former Professor Indian Institute of Technology Bombay Delhi-110092
MECHANICS OF MATERIALS
CHATR Stress MCHANICS OF MATRIALS and Strain Axial Loading Stress & Strain: Axial Loading Suitability of a structure or machine may depend on the deformations in the structure as well as the stresses induced
BUCKLING COEFFICIENTS FOR SANDWICH CYLINDERS OF FINITE LENGTH UNDER UNIFORM EXTERNAL LATERAL PRESSURE
U.S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY MADlSON, WIS. In Cooperation with the University of Wisconsin U. S. FOREST SERVICE RESEARCH NOTE FPL-0104 SEPTEMBER 1965 BUCKLING
[5] Stress and Strain
![[5] Stress and Strain](/thumbs/95/123344550.jpg "[5] Stress and Strain")
[5] Stress and Strain Page 1 of 34 [5] Stress and Strain [5.1] Internal Stress of Solids [5.2] Design of Simple Connections (will not be covered in class) [5.3] Deformation and Strain [5.4] Hooke s Law
Comb resonator design (2)
Lecture 6: Comb resonator design () -Intro Intro. to Mechanics of Materials School of Electrical l Engineering i and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory
Chapter 2: Elasticity
OHP 1 Mechanical Properties of Materials Chapter 2: lasticity Prof. Wenjea J. Tseng ( 曾文甲 ) Department of Materials ngineering National Chung Hsing University wenjea@dragon.nchu.edu.tw Reference: W.F.
Unit M1.5 Statically Indeterminate Systems
Unit M1.5 Statically Indeterminate Systems Readings: CDL 2.1, 2.3, 2.4, 2.7 16.001/002 -- Unified Engineering Department of Aeronautics and Astronautics Massachusetts Institute of Technology LEARNING OBJECTIVES
7 TRANSVERSE SHEAR transverse shear stress longitudinal shear stresses
7 TRANSVERSE SHEAR Before we develop a relationship that describes the shear-stress distribution over the cross section of a beam, we will make some preliminary remarks regarding the way shear acts within
AND RELATED PROBLEMS BENDING OF AN INCOMPLETE ANNULAR PLATE. J. R. BARBER The University of' Newcastle upon Tyne. Member of ihe Institution
BENDING OF AN INCOMPLETE ANNULAR PLATE AND RELATED PROBLEMS J. R. BARBER The University of' Newcastle upon Tyne. Member of ihe Institution Closed-form solutions and stress-concentration data are obtained
a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
A L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
Partial Differential Equations (PDEs)
C H A P T E R Partial Differential Equations (PDEs) 5 A PDE is an equation that contains one or more partial derivatives of an unknown function that depends on at least two variables. Usually one of these
Interstate 35W Bridge Collapse in Minnesota (2007) AP Photo/Pioneer Press, Brandi Jade Thomas
7 Interstate 35W Bridge Collapse in Minnesota (2007) AP Photo/Pioneer Press, Brandi Jade Thomas Deflections of Trusses, Beams, and Frames: Work Energy Methods 7.1 Work 7.2 Principle of Virtual Work 7.3
Basic concepts to start Mechanics of Materials
Basic concepts to start Mechanics of Materials Georges Cailletaud Centre des Matériaux Ecole des Mines de Paris/CNRS Notations Notations (maths) (1/2) A vector v (element of a vectorial space) can be seen
SKAA 1213 Engineering Mechanics
SKAA 1213 Engineering Mechanics TPIC 5 Moment and Couple Lecturers: Rosli Anang Dr. Mohd Yunus Ishak Dr. Tan Cher Siang Moment of a Force Moment of a force about a point/axis the tendency of the force
PILE SOIL INTERACTION MOMENT AREA METHOD
Pile IGC Soil 2009, Interaction Moment Guntur, INDIA Area Method PILE SOIL INTERACTION MOMENT AREA METHOD D.M. Dewaikar Professor, Department of Civil Engineering, IIT Bombay, Mumbai 400 076, India. E-mail:
External Work. When a force F undergoes a displacement dx in the same direction i as the force, the work done is
Structure Analysis I Chapter 9 Deflection Energy Method External Work Energy Method When a force F undergoes a displacement dx in the same direction i as the force, the work done is du e = F dx If the
Stress and Strain ( , 3.14) MAE 316 Strength of Mechanical Components NC State University Department of Mechanical & Aerospace Engineering
(3.8-3.1, 3.14) MAE 316 Strength of Mechanical Components NC State Universit Department of Mechanical & Aerospace Engineering 1 Introduction MAE 316 is a continuation of MAE 314 (solid mechanics) Review
Engineering Tripos Part IIA Supervisor Version. Module 3D4 Structural Analysis and Stability Handout 1
Engineering Tripos Part A Supervisor Version Module 3D4 Structural Analysis and Stability Handout 1 Elastic Analysis (8 Lectures) Fehmi Cirak (fc86@) Stability (8 Lectures) Allan McRobie (fam @eng) January
ELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 998 Comments to the author at krm@mathsuqeduau All contents copyright c 99 Keith
Module III - Macro-mechanics of Lamina. Lecture 23. Macro-Mechanics of Lamina
Module III - Macro-mechanics of Lamina Lecture 23 Macro-Mechanics of Lamina For better understanding of the macromechanics of lamina, the knowledge of the material properties in essential. Therefore, the
ELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS
SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by
SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e
N = N A Pb A Pb. = ln N Q v kt. = kt ln v N
5. Calculate the energy for vacancy formation in silver, given that the equilibrium number of vacancies at 800 C (1073 K) is 3.6 10 3 m 3. The atomic weight and density (at 800 C) for silver are, respectively,
Executive Committee and Officers ( )
Gifted and Talented International V o l u m e 2 4, N u m b e r 2, D e c e m b e r, 2 0 0 9. G i f t e d a n d T a l e n t e d I n t e r n a t i o n a2 l 4 ( 2), D e c e m b e r, 2 0 0 9. 1 T h e W o r
Chapter 3 LAMINATED MODEL DERIVATION
17 Chapter 3 LAMINATED MODEL DERIVATION 3.1 Fundamental Poisson Equation The simplest version of the frictionless laminated model was originally introduced in 1961 by Salamon, and more recently explored
HIGHER-ORDER THEORIES
HIGHER-ORDER THEORIES THIRD-ORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1 Third-Order Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) +
ELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis
uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods
Quintic beam closed form matrices (revised 2/21, 2/23/12) General elastic beam with an elastic foundation
General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation
A study of the critical condition of a battened column and a frame by classical methods
University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 003 A study of the critical condition of a battened column and a frame by classical methods Jamal A.H Bekdache
Branch: Name of the Student: Unit I (Fourier Series) Fourier Series in the interval (0,2 l) Engineering Mathematics Material SUBJECT NAME
13 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE UPDATED ON : Transforms and Partial Differential Equation : MA11 : University Questions :SKMA13 : May June 13 Name of the Student: Branch: Unit
Partial Differential Equations
Partial Differential Equations Xu Chen Assistant Professor United Technologies Engineering Build, Rm. 382 Department of Mechanical Engineering University of Connecticut xchen@engr.uconn.edu Contents 1
Flow-Induced Vibration of Pipeline on Elastic Support
Available online at www.sciencedirect.com Procedia Engineering () 986 99 The Twelfth East Asia-Pacific Conference on Structural Engineering and Construction Flow-Induced Vibration of Pipeline on Elastic
176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s
A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps
4.MECHANICAL PROPERTIES OF MATERIALS
4.MECHANICAL PROPERTIES OF MATERIALS The diagram representing the relation between stress and strain in a given material is an important characteristic of the material. To obtain the stress-strain diagram
Lab Exercise #5: Tension and Bending with Strain Gages
Lab Exercise #5: Tension and Bending with Strain Gages Pre-lab assignment: Yes No Goals: 1. To evaluate tension and bending stress models and Hooke s Law. a. σ = Mc/I and σ = P/A 2. To determine material
Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor
Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor Somnath Bhowmick Materials Science and Engineering, IIT Kanpur April 6, 2018 Tensile test and Hooke s Law Upto certain strain (0.75),
UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES
UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES A Thesis by WOORAM KIM Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the
14. *14.8 CASTIGLIANO S THEOREM
*14.8 CASTIGLIANO S THEOREM Consider a body of arbitrary shape subjected to a series of n forces P 1, P 2, P n. Since external work done by forces is equal to internal strain energy stored in body, by
ENG2000 Chapter 7 Beams. ENG2000: R.I. Hornsey Beam: 1
ENG2000 Chapter 7 Beams ENG2000: R.I. Hornsey Beam: 1 Overview In this chapter, we consider the stresses and moments present in loaded beams shear stress and bending moment diagrams We will also look at